Algebra Examples

$- \frac{1}{2} {(x - 3)}^{2} = 12$

Step 1

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{1}{2} \cdot {(x - 3)}^{2}) = - 2 \cdot 12$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $- 2 (- \frac{1}{2} \cdot {(x - 3)}^{2})$ .

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Step 2.1.1.1

Combine ${(x - 3)}^{2}$ and $\frac{1}{2}$ .

$- 2 (- \frac{{(x - 3)}^{2}}{2}) = - 2 \cdot 12$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Move the leading negative in $- \frac{{(x - 3)}^{2}}{2}$ into the numerator.

$- 2 \frac{- {(x - 3)}^{2}}{2} = - 2 \cdot 12$

Step 2.1.1.2.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- {(x - 3)}^{2}}{2} = - 2 \cdot 12$

Step 2.1.1.2.3

Cancel the common factor.

$2 \cdot - 1 \frac{- {(x - 3)}^{2}}{2} = - 2 \cdot 12$

Step 2.1.1.2.4

Rewrite the expression.

$- - {(x - 3)}^{2} = - 2 \cdot 12$

Step 2.1.1.3

Multiply.

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Step 2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 {(x - 3)}^{2} = - 2 \cdot 12$

Step 2.1.1.3.2

Multiply ${(x - 3)}^{2}$ by $1$ .

${(x - 3)}^{2} = - 2 \cdot 12$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 2$ by $12$ .

${(x - 3)}^{2} = - 24$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 3 = \pm \sqrt{- 24}$

Step 4

Simplify $\pm \sqrt{- 24}$ .

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Step 4.1

Rewrite $- 24$ as $- 1 (24)$ .

$x - 3 = \pm \sqrt{- 1 (24)}$

Step 4.2

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x - 3 = \pm \sqrt{- 1} \cdot \sqrt{24}$

Step 4.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x - 3 = \pm i \cdot \sqrt{24}$

Step 4.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 4.4.1

Factor $4$ out of $24$ .

$x - 3 = \pm i \cdot \sqrt{4 (6)}$

Step 4.4.2

Rewrite $4$ as $2^{2}$ .

$x - 3 = \pm i \cdot \sqrt{2^{2} \cdot 6}$

Step 4.5

Pull terms out from under the radical.

$x - 3 = \pm i \cdot (2 \sqrt{6})$

Step 4.6

Move $2$ to the left of $i$ .

$x - 3 = \pm 2 i \sqrt{6}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 3 = 2 i \sqrt{6}$

Step 5.2

Add $3$ to both sides of the equation.

$x = 2 i \sqrt{6} + 3$

Step 5.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 3 = - 2 i \sqrt{6}$

Step 5.4

Add $3$ to both sides of the equation.

$x = - 2 i \sqrt{6} + 3$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i \sqrt{6} + 3, - 2 i \sqrt{6} + 3$